All these wormhole trajectories are plotted on figure REF . We checked numerically that at a throat, \(P\rightarrow -1\)  and \(\beta ^{\prime }\rightarrow 1+1/w\) . None of them tends to the Minkowski spacetime[1]} since it is a saddle point whatever \(w\) , as indicated in table REF . Concerning the presence of singularities, since \(P=-1\)  at the throat, \(\operatorname{arctanh}{\beta }\operatorname{arctanh}{\theta }\)  is vanishing and there is none there. All the trajectories end in \(u\rightarrow \infty \) , without reaching the lines \(\theta =\pm 1\)  or \(\beta =\pm 1\)  at finite \(u\) . The density and the scalars are thus finite everywhere at finite \(u\)  and these wormhole solutions are free from singularity.
